Modified Iterative Method for Multiple Roots

Abstract

     Boole's rule and Weddle's rule modified quadrature iterated techniques are provided in this study for locating multiple non-linear equations' roots, the suggested approaches converged cubically, Newton's approach was used for discovering numerous nonlinear equation roots. Several modifications have been made to achieve a higher degree of convergence. The modified classical methods developed by many authors to solve multiple roots of nonlinear equations have been effective in overcoming the deficiency of the classical Newton Raphson method, however there are new trends of methods proposed by authors, which have proven to be more efficient than some already existing ones. There are several numerical examples that support the suggested technique's justification as an evaluation of the Newton-Raphson method and Simpson's approach, Maple 18 is used to investigate numerical result representations of modified quadrature iterative Algorithms. The result from numerical findings is that the presented modified quadrature iterated techniques for finding multiple roots of non-linear equations outperform existing approaches in terms of performance; these methods were programmed by using software Maple.

Background: Nonlinear equations with multiple roots are challenging to solve accurately using classical Newton’s method due to its slow convergence in such cases. This limitation motivates the development of improved iterative methods that offer faster and more reliable convergence. These enhanced techniques aim to overcome the shortcomings of traditional approaches.

Results: This study developed enhanced iterative methods to improve Newton’s method for solving equations with multiple roots. The new techniques proved to be accurate, fast, and reliable through various practical tests. Using Maple 18 for implementation, these methods demonstrated strong performance compared to existing approaches.

Conclusion This work presented improved iterative methods that make Newton’s technique more effective for equations with multiple roots. Through several practical tests, the new methods proved to be accurate, fast, and reliable. Built and tested using Maple 18, they held up well against other well-known approaches